100 Years of Relativity Space-Time Structure: Einstein and Beyond ...

324 T. Banksform:KMV = e P 2 [K i¯j D i W D¯j ¯W − 3M 2 P|W | 2 ].The would be moduli space is a Kahler manifold with Kahler potential K.The superpotential, W is a section of a holomorphic line bundle over thismanifold, and D i = ∂ i − ∂iK is the covariant derivative on this bundle.MP2Generically, there will be solutions of D i W = 0, which is the condition topreserve SUSY, and also guarantees that we are at a stationary point ofthe potential. However, W will vanish at these points only in exceptionalcases. This analysis leads us to expect supersymmetric theories with AdSasymptotics and and non-supersymmetric theories with any value of thecosmological constant, but zero seems unlikely. Indeed, attempts to breaksupersymmetry in supergravity and perturbative string theory sometimespreserve vanishing c.c. to one or more orders of perturbation theory, but noone has found an example with broken SUSY and exactly vanishing c.c. .This has led me to conjecture that there are no Poincare invariant theoriesof quantum gravity, which are not Super-Poincare invariant.We will have more to say about the validity of this conjecture below.Next however, we will see what we can learn about theories of quantumgravity with AdS asymptotics.5. The AdS/CFT CorrespondenceThe AdS/CFT correspondence came out of the analysis of black hole entropyin string theory. Susskind 20 was the first to suggest that the exponentialdegeneracies of states found in perturbative string theory were relatedto black hole entropy l . Sen pointed out that the arguments could be mademore precise by considering extremal black holes which satisfied the BPScondition 22 . The idea was that BPS degeneracies can be counted in a weakcoupling approximation, in which the gravitational effects that make thestates into black holes are neglected. The BPS property ensures that themasses and degeneracies are independent of coupling. Sen studied blackholes with zero classical horizon area. By computing instead the area of a“stretched horizon” a few string lengths from the classical horizon, he foundl This suggestion seemed a bit obscure since the powers of energy in the exponentialdid not match, but it was validated by the correspondence principle of Horowitz andPolchinski 21 .